We prove that there is a polyhedron with genus 6 whose faces are
orthogonal polygons (equivalently, rectangles) and yet the angles between some
faces are not multiples of 90°, so the polyhedron itself is not
orthogonal. On the other hand, we prove that any such polyhedron must have
genus at least 3. These results improve the bounds of Donoso and O'Rourke
[4] that there are nonorthogonal polyhedra with orthogonal faces and
genus 7 or larger, and any such polyhedron must have genus at
least 2. We also demonstrate nonoverlapping one-piece edge-unfoldings
(nets) for the genus-7 and genus-6 polyhedra.